Deterministic Silicon Pillar Assemblies and their Photonic Applications
Bikash Dev Choudhury
Doctoral thesis in Physics Stockholm, Sweden 2016
Division of Semiconductor Materials
School of Information and Communication Technology KTH Royal Institute of Technology
Deterministic Silicon Pillar Assemblies and their Photonic Applications
TRITA-ICT 2016:06 ISBN 978-91-7595-891-0
Akademisk avhandling som med tillstånd av Kungl Tekniska högskolan framlägges till offentlig granskning för avläggande av teknologie doktorsexamen i fysik fredag den 29 april 2016 klockan 10.00 i sal 208, Electrum KTH Skolan för informations- och kommunikationsteknik, Kistagången 16,Kista, Sverige.
© Bikash Dev Choudhury, April 2016 Printed by US-AB, Stockholm, 2016
Cover picture: The left panel top image (SEM) shows tapered Si nanopillars array, fabricated by colloidal lithography and dry etching and the bottom image is corresponding second-harmonic generation intensity graph vs wavelength. In the middle panel, top image is ZnO nanowire grown hierarchically on Si micropyramid arrays by hydrothermal process and bottom image is its total reflectivity graph vs wavelength in comparison to planar Si. The right panel top image (SEM) is showing vertical Si nanopillars in an aperiodic arrangement fabricated by nanoimprint lithography and dry etching for color filter applications and bottom picture shows reflected and transmitted light by the nanopillar color filters.
KTH School of Information and communication Technology
SE-16440, KISTA Sweden
Table of Contents
Abstract ... v
Abstract (Swedish) ... vii
Acknowledgements ... ix
List of papers ... xi
Acronyms ... xiii
1 Introduction ... 1
1.1 Si micro/nano structures for antireflection and photovoltaics ... 2
1.2 Si nanopillar arrays for optical biosensing ... 4
1.3 Surface Second Harmonic (SHG) generation in Si nanopillar arrays ... 4
1.4 Aperiodic Si nanopillar assemblies for color filtering applications ... 5
1.5 Thesis outline ... 6
2 Background and Theory ... 7
2.1 Optical properties of Silicon ... 7
2.2 Optical properties of nanopillars ... 8
2.3 Antireflection property and solar cell ... 10
2.4 Radial junction (pillar) solar cell ... 13
2.5 Solar spectrum ... 13
2.6 Nanopillars and optical biosensing ... 14
2.7 Second harmonic generation and Si nanopillars ... 16
2.8 Photonic nano structures and color filters ... 18
2.8.1 Human eye and color vision ... 18
2.8.2 Colors in nature ... 19
2.8.3 Quantitative description of color ... 21
2.8.4 Nanophotonic color filter ... 22
2.8.5 Transverse localization effect in aperiodic nanopillar assemblies ... 23
2.9 Numerical methods ... 24
3 Experimental Methods ... 29
3.1 Fabrication Methodology ... 29
3.1.1 Colloidal lithography ... 29
3.1.2 Nanoimprint lithography ... 30
3.1.3 Dry etching of Si ... 31
3.1.4 Pseudo Bosch process ... 32
3.2 Optical measurement and material characterization methods ... 34
3.2.1 UV-visible spectrophotometer measurements ... 34
4 Antireflection and Photovoltaic Applications ... 37
4.1 Silicon micro-structure and ZnO nano-wire hierarchical assortments ... 37
4.1.1 Fabrication processes ... 37
4.1.2 Measurement ... 38
4.1.3 Conclusion ... 39
4.2 RTA processed radial junction Si nanopillar solar cell ... 39
4.2.1 Nanopillar Si solar cell ... 40
4.2.1 Characterization ... 41
4.2.1 Conclusion ... 42
5 Nanopillar Optical Biosensing ... 43
5.1 Si nanopillar array biosensors ... 43
5.2 Biosensor responses ... 43
5.3 Optical RI sensing by Silicon nanopillar Arrays ... 45
5.4 Conclusion ... 48
6 Surface Second Harmonic Generation with Vertical Silicon Pillar ... 49
6.1 Experimental methods ... 49
6.2 FDTD Simulation ... 50
6.3 Conclusion ... 50
7 Nanopillar Color Filter ... 51
7.1 Design and fabrication of the nanopillar filter ... 52
7.2 Transmission characteristics of the color filters ... 53
7.3 Transmission spectral characteristics: Angle, polarization and spatial tuning ... 55
7.4 Transverse light localization in aperiodic NP assemblies ... 56
7.5 Conclusion ... 58
8 Summary and Outlook... 59
Bibliography ... 63
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Abstract
It is of paramount importance to our society that the environment, life style, science and amusement flourish together in a balanced way. Some trends in this direction are the increased utilization of renewable energy, like solar photovoltaics; better health care products, for example advanced biosensors; high definition TV or high resolution cameras; and novel scientific tools for better understanding of scientific observations. Advancement of micro and nanotechnologies has directly and positively impacted our stance in these application domains; one example is that of vertical periodic or aperiodic nano or micro pillar assemblies which have attracted significant research and industrial interest in recent years. In particular, Si pillars are very attractive due to the versatility of silicon. There are many potential applications of Si nanopillar/nanowire assemblies ranging from light emission, solar cells, antireflection, sensing and nonlinear optical effects. Compared to bulk, Si pillars or their assemblies have several unique properties, such as high surface to volume ratios, light localization, efficient light guiding, better light absorption, selective band of light propagation etc.
The focus of the thesis is on the fabrication of Si pillar assemblies and hierarchical ZnO nanowires on Si micro structures in top-down and bottom-up approaches and their optical properties and different applications. Here, we have investigated periodic and aperiodic Si nano and micro structure assemblies and their properties, such as light propagation, localization, and selective guiding and light- matter interaction. These properties are exploited in a few important optoelectronic/photonic applications, such as optical biosensors, broad-band anti-reflection, radial-junction solar cells, second harmonic generation and color filters.
We achieved a low average reflectivity of ~ 2.5 % with the periodic Si micropyramid-ZnO NWs hierarchical arrays. Tenfold enhancement in Raman intensity is also observed in these structures compared to planar Si. These Si microstructure-ZnO NW hierarchical structures can enhance the performance and versatility of photovoltaic devices and optical sensors. A convenient top-down fabrication of radial junction nanopillar solar cell using spin-on doping and rapid thermal annealing process is presented. Broad band suppressed reflection, on average 5%, in 300- 850 nm wavelength range and an un-optimized cell efficiency of 6.2 % are achieved. Our method can lead to a simple and low cost process for high efficiency radial junction nanopillar solar cell fabrication.
Silicon dioxide (SiO2) coated silicon nanopillar (NP) arrays are demonstrated for surface sensitive optical biosensing. Bovine serum albumin (BSA)/anti-BSA model system is used for biosensing trials by photo-spectrometry in reflection mode. Best sensitivity in terms of limit of detection of 5.2 ng/ml is determined for our nanopillar biosensor. These results are promising for surface sensitive biosensors and the technology allows integration in the CMOS platform.
Si pillar arrays used for surface second harmonic generation (SHG) experiments are shown to have a strong dependence of the SHG intensity on the pillar geometry. The surface SHG can be suitable for nonlinear silicon photonics, surface/interface studies and optical sensing.
Aperiodic Si nanopillar assemblies in PDMS matrix are demonstrated for efficient color filtering in transmission mode. These assemblies are designed using the ‘‘molecular dynamics-collision between hard sphere’’ algorithm. The designed structure is modeled in a 3D finite difference time domain (FDTD) simulation tool for optimization of color filtering properties. Transverse localization effect of light in our nanopillar color filter structures is investigated theoretically and the results are very promising to achieve image sensors with high pixel densities (~1 µm) and low crosstalk. The developed color filter is applicable as a stand-alone filter for visible color in its present form and can be adapted for displays, imaging, smart windows and aesthetic applications.
Keywords: nanopillar, nanowires, nanophotonics, nanofabrication, silicon, photovoltaics, second-harmonic generation, top-down approach, colloidal lithography, color filter, biosensor
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Deterministisk Silicon pelare församlingar och deras fotonik användningar
Sammanfattning
Det är av största vikt för vårt samhälle att miljö, livsstil, vetenskap och nöje blomstrar tillsammans på ett balanserat sätt. Några trender i denna riktning är den ökade användningen av förnybar energi, som solceller, bättre sjukvårdsprodukter, till exempel avancerade biosensorer; HD TV eller högupplösta kameror; och nya vetenskapliga verktyg för bättre förståelse för vetenskapliga observationer. Framsteg inom mikro- och nanoteknik har direkt och positivt påverkat vår ställning inom dessa tillämpningsområden; ett exempel är vertikala periodiska och icke-periodiska nano- och mikropelarstrukturer som har lockat betydande intresse inom forskning och industri under de senaste åren. I synnerhet kiselpelare är mycket attraktiva på grund av mångsidigheten hos kisel. Det finns många potentiella tillämpningar av kisel-nanopelarstrukturer/-nanotrådstrukturer som sträcker sig från ljusemission, solceller, antireflex till avkänning och icke-linjära optiska effekter. Jämfört med bulkmaterial har kiselpelare och deras strukturer flera unika egenskaper, såsom hög yta per volym, ljuslokalisering, effektiv ljusledning, bättre ljusabsorption, bandselektivitet för ljusutbredning etc.
Fokus i avhandlingen ligger på tillverkning av kiselpelarstrukturer och hierarkiska ZnO-nanotrådar på kiselmikrostrukturer med top-down och bottom-up-metoder och deras optiska egenskaper och olika applikationer. Här har vi undersökt periodiska och icke-periodiska kiselnano- och kiselmikro- strukturer och deras egenskaper, såsom ljusutbredning, lokalisering, och selektiv styrning och ljus- materia-växelverkan. Dessa egenskaper utnyttjas i några viktiga optoelektroniska/fotoniska tillämpningar, såsom optiska biosensorer, bredbandigt antireflex, radiell-övergångs-solceller och frekvensdubbling och färgfilter.
Vi uppnådde en låg genomsnittlig reflektivitet av ~2,5% med periodiska kiselmikropyramid-ZnO- nanotrådar hierarkiska uppställningar. Tiofaldig ökning i Ramanintensitet observeras också i dessa strukturer jämfört med plant kisel. Dessa hierarkiska kiselmikrostruktur-ZnO-nanotrådstrukturer kan förbättra prestandan och mångsidigheten hos fotovoltaiska enheter och optiska sensorer. En praktisk top-down tillverkning av radiell-övergångs-nanopelar-solcell med hjälp av spin-on dopning och snabb termisk glödgningsprocess presenteras. Bredbandigt dämpad reflektion på i genomsnitt 5% i våglängdsområdet 300-850nm och en icke-optimerad cellverkningsgrad på 6,2% uppnås. Vår metod kan leda till en enkel och billig process för tillverkning av högeffektiva radiell-övergångs-nanopelar- solceller.
Kiseldioxid(SiO2)-belagda kiselnanopelar(NP)-uppställningar demonstreras för ytkänslig optisk bioavkänning. Bovint serumalbumin-(BSA)/anti-BSA-modellsystem används för bioavkänningsprövningar genom fotospektrometri i reflektionsmod. Bästa känslighet i termer av detektionsgräns på 5,2 ng/ml uppnås för vår nanopelar-biosensor. Dessa resultat är lovande för ytkänsliga biosensorer och tekniken möjliggör CMOS-integrering.
Si-pelaruppställningar som används för experiment med frekvensdubbling i ytskiktet har visat sig ha ett starkt beroende av frekvensdubblingsintensiteten över pelargeometrin. Ytskiktsfrekvensdubbling kan vara lämpligt för icke-linjär kiselfotonik, yt- och gränssnittsundersöknignar och optisk avkänning.
Icke-periodiska kiselnanopelarstrukturer i PDMS-matris demonstreras för effektiv färgfiltrering i sändningsläge. Dessa enheter är utformade med hjälp av ''molekyldynamik-kollision mellan hårda sfärer''-algoritmen. Den utformade strukturen modelleras i ett simuleringsverktyg för optimering av färgfiltreringsegenskaper som använder 3D-finita tidsdifferensdomänen (FDTD). Den transversella lokaliseringen av ljus i våra nanopelar-färgfilterstrukturer undersöks teoretiskt och resultaten är mycket lovande för att uppnå bildsensorer med hög pixeltäthet (~1pm) och låg överhörning. Det utvecklade färgfiltret är tillämpligt som ett fristående filter för synlig färg i sin nuvarande form och kan anpassas för visning, avbildningssmarta fönster och konstnärliga tillämpningar.
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Acknowledgements
Retrospection of last few years of my research brings to my mind the names of all those who have guided, helped and encouraged me to accomplish my thesis. I am truly indebted to all of them beyond narratives and altitudes.
First of all, I would like to thank my wife, Ivy and my little gem, Adhrit for being there with me, supporting and motivating all the way through. I have really taken both of you along the toughest ride of doctoral research and still you stand by my side, big applause.
I am most grateful to my brilliant supervisor, Prof. Srinivasan Anand, for giving me the opportunity to conduct research in his group under his able guidance. I sincerely appreciate his support and encouragement to become a better researcher and deliver as an entrepreneur.
His endurance, encouragement and guidance on my work helped in augmenting the work both scientifically and for applicability. He has been a great support throughout be it academic or personal matter. Thank you a lot Anand!
Next, I would like to express my gratitude to co-supervisor Prof. Sebastian Lourdudoss, head of the Material and Nanophysics department, for his extensive support. His friendliness and welcoming attitude have been very helpful and encouraging.
Special appreciation to Dr. Marcin Swillo for his support and for sharing his knowledge on optics and photonics. I have learned quite a lot from him. Especially while working with him, I have developed an interest in exploring possibilities of further application of second harmonic generation from Silicon.
I would not have embarked on the journey of my doctoral research had it not been for Dr. K.
Chalapathi, former HOD, optoelectronics division SAMEER, Mumbai. Sir has ignited the sprit in me to go for further research. A heartfelt gratitude!
Pankaj Kumar Sahoo, PhD scholar, IIT Delhi deserves particular mention for his time in our Department. My research got another dimension when I and Pankaj started working together on ‘color filter’. The period we worked together is the most interesting and eventful phase of my research.
A sincere word of gratitude goes out to all my colleagues with whom I had the pleasure to work in the lab and collaborated on different projects. I should thank Dr. Reza Sanatinia, Dr.
Carl Junesand, Dr. Shagufta Naureen, Dr. Naeem Shahid, Dr. Himanshu Kataria, Dr. Apurba Dev, Dr. Wondwosen Metaferia, Ahmad Abedin, Dennis Visser, Dr. Oruganty Satya Murthy, Giriprasanth Omanakuttan and Joakim Storck.
During my PhD studies I have got the opportunity to partially guide Mohamed Saad, Evert Ebraert, and IneseKrasovska for their master thesis. I value it as a rewarding experience and thank all of them. I truly appreciate Mohamed Saad for his software related help.
I am thankful to Ms. Madeleine Printzsköld for her constant care and continued support with all the administrative issues. She certainly makes life much easier for PhD students in our department.
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I would like to acknowledge the funding from the Swedish Research Council VR and the FP7 EU network of excellence, Nanophotonics for Energy Efficiency (N4E), NANORDSUN (funded by Nordic Innovation center), and Linné Center for Advanced Optics and Photonics (ADOPT).
My Ph.D. thesis work was partially funded by Erasmus Mundus Action-2 project EurIndia scholarship and I am duly grateful to all concern. Specially, I would like to thank Alphonsa Lourdudoss for facilitating the grant and her support throughout my tenure.
My extended family; brothers and sisters deserve special gratitude for their support.
Lastly, I am indebted to my parents who had showered illimitable affection and blessing on me.
Bikash Dev Choudhury Stockholm, April 2016
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List of papers
Papers included in this thesis and contribution
I. Bikash Dev Choudhury, Ahmad Abedin, Apurba Dev, Reza Sanatinia and Srinivasan Anand, ''Silicon micro-structure and ZnO nanowire hierarchical assortments for light management,'' Optical Materials Express, ISSN: 2159- 3930, Vol. 3, No. 8, pp. 1039-1048(2013).
[Major part of writing, Si pillar fabrication, sample preparation for ZnO growth and major part of optical measurements]
II. Bikash Dev Choudhury and Srinivasan Anand, ‘‘RTA treated spin–on doped antireflective radial junction Si nanopillar solar cell’’, Manuscript.
[Major part of writing, processing, device fabrication and characterization, simulations and analysis]
III. B. Dev Choudhury, R. Casquel, M.J. Bañuls, F.J. Sanza, M.F. Laguna, M.
Holgado, R. Puchades, A. Maquieira, C.A. Barrios and S. Anand, ''Silicon nanopillar arrays with SiO2 overlayer for biosensing application,'' Optical Materials Express, Vol. 4, No. 7,pp.1345-1354 (2014).
[Major part of writing, fabrication of Si pillar arrays, fabrication of planar test samples for bio-functionalization experiments, electromagnetic simulations and part of analysis]
IV. B. Dev Choudhury, Pankaj K. Sahoo, R. Sanatinia, Guillermo Andler, S.
Anand and M. Swillo, ''Surface second harmonic generation from silicon pillar arrays with strong geometrical dependence'', Optics letters, Vol. 40, No. 9 ,pp. 2072-2075(2015).
[Major part of writing, sample fabrication, participated in the measurements, major part of electromagnetic simulations and associated analysis]
V. B. Dev Choudhury, Pankaj K. Sahoo and Srinivasan Anand, ‘‘Nanopillar Assemblies with Deterministic Correlated Disorder for Color Filtering’’, Manuscript.
[Proposed and adapted the existing (in the group) embedded semiconductor nanopillar concepts/technologies for color filter applications; Major part of writing, major part of fabrication, measurements, electromagnetic simulations and analysis]
VI. D. Visser, B. Dev Choudhury, I. Krasovska and S. Anand, ‘‘Optical RI Sensing in the Visible/NIR Spectrum by Silicon Nanopillar Arrays’’, Manuscript.
[Si pillar array fabrication, part of electromagnetic simulations, part of analysis, minor part in writing]
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Other relevant contributions not included in this thesis Patent application:
1. ‘‘An Optical Transmission Filter”; Applicants: Bikash Dev
Choudhury,Anand Srinivasan and Pankaj Kr. Sahoo, Swedish patent Application number 1550804-7.
Journal paper:
1. A Dev, B. Dev Choudhury, A. Abedin and S. Anand, ‘Fabrication of Periodic Nanostructure Assemblies by Interfacial Energy Driven Colloidal Lithography'', Advanced Functional Materials, Vol. 24, Issue 29, PP. 4577–4583(2014).
Conference contribution:
1. Bikash Dev Choudhury, Evert Ebraert, Srinivasan Anand, ''Optimization of light absorption in semiconductor nano-pillar array solar cells'',abstract, TOM3_6008_014, European optical society annual meetting(EOSAM),25th-28th September,2012,Aberdeen ,Scotland, UK.
2. R. Sanatinia,K. M. Awan, S. Naureen, E. Ebraert, B. D. Choudhury, S.
Anand,''Fabrication of GaAs nanopillars with optimized design for enhanced sunlight absorption'',abstract,TOM3_6032_040,European optical society annual meeting (EOSAM),25th-28th September,2012, Aberdeen, Scotland, UK.
3. Apurba Dev, Bikash Dev Choudhury, Shagufta Naureen, Naeem Shahid and Srinivasan Anand,''Spatially resolved characterization of InP-based nanopillar arrays by scanning capacitance microscopy'',abstract, European material research society(E-MRS) Fall meeting,17th-21st September, 2012,Warsaw, Poland.
4. S. Naureen, R. Sanatinia, B. Choudhury, N. Shahid, R. Perumal and S.
Anand, ''Nanopattering of semiconductors using self-assembled silicon- di-oxide nanospheres as etch masks'',Proceedings Semiconductor and insulating material conference (SIMC-XVI), Tu3-6,19th- 23rd June,2011, Stockholm, Sweden.
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Acronyms
EMT effective medium theory
EMA effective medium approximation ARC antireflection coating
CMOS complementary metal-oxide semiconductor DL limit of detection, (biosensor context) ICP inductively coupled plasma
I-V current-voltage
FDTD finite difference time domain
FEM finite element method
FWHM full width at half maximum ICP inductively coupled plasma
IR infrared
ITO indium tin oxide
I-V current-voltage
ICP-RIE inductively coupled plasma reactive ion etching IPF initial packing fraction
LED light emitting diode
NIR near-infrared
NP nanopillar
NW nanowire
PDMS polydimethylsiloxane
PV photovoltaic
RGB red, green and blue, (color filter context)
RI refractive index
RIE reactive ion etching
RTA rapid thermal annealing
SOD spin-on doping
SEM scanning electron microscope SHG second –harmonic generation
UV ultraviolet
ZnO zinc oxide
1
1 Introduction
Silicon is one of the most versatile semiconductor materials for electronics and photonic applications [1, 2]. In fact, there has been a strong emphasis on incorporating active optoelectronic functionalities in silicon technology. In the last few decades there has been lot of research on the utilization of low dimensional Si structures for photonic applications [3].
While migrating from bulk to nano-/sub-micron structures, several material properties also change. One obvious change is in its geometrical appearance in terms of mechanical transformations. Similarly, optical, electrical, transport and thermal properties of nanostructures deviate from bulk [4]. In addition, depending on the size of the structures quantum mechanical behavior can dominate their electronic properties. There has been extensive research to understand the fascinating properties of nano or micro structures and their utilization in a variety of applications ranging from electronics, photonics, renewable energy, security to health care, to meet societal needs [5]. In addition to human innovation, many nano or micro structure related phenomena (e.g. colors, hydrophobicity etc.) are found in nature and have inspired researchers to adapt such functional structures, so called biomimetic or bioinspired structures, in device applications.
If we look at nature, there is a widespread of structural colors mostly generated by light scattering, interference and diffraction by visible wavelength scale nanostructures [6-9]. In several cases, the arrangement of such structures is periodic, more like photonic crystals, and vary in refractive index either one, two or three dimensions. However, the structures can have long range periodicity or short range periodicity or a completely random arrangement, with each case giving rise to different observational optical effects for specific purposes.
There are several birds, butterflies, beetles, and marine animals that exploit periodic photonic nanostructures on their surfaces to change color with viewing angle (iridescence) [10]. On the other hand, the non-iridescent structural colors of the feathers of some birds (e.g.
male plum-throated Cotinga) are generally produced by three-dimensional, quasi-ordered nanostructures [11]. On a different note, the dome shaped structures found in ‘moth eyes’, the reason for its dark eyes, is an example of antireflective structures in nature [12]. Such examples from nature, in a way, can act as guiding principles for many practical applications.
In a laboratory environment, nanostructure or micro structure assemblies are fabricated and utilized for many applications ranging from antireflection, sensing, color imaging etc.
Broadly speaking, there are two ways to realize these kinds of structures: a top down approach, where one typically starts with a planar substrate and etch it down with particular
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patterns to achieve assemblies of nano/microstructures [13-16]; and a bottom up approach, where one builds the nano/microstructure assemblies on a substrate by growth/deposition methods [17]. However, combinations of both approaches are also often used.
Vertical nanostructures, in particular periodic nanopillars arrays have generated lot of industrial and academic interest for their unique properties [18]. Vertical Si nano structures (nanopillar or nanowire) are used for generation of light, sensing applications, solar cells, photo detectors etc. [19-21]. Recently, efforts on aperiodic assemblies of nanostructures are also gaining momentum in fundamental and applied research in nanophotonics. The utility of aperiodic structures has been reported for solar photovoltaics and for antireflection, and useful phenomena such as light localization have been observed which is absent in periodic systems [22-26].
This thesis investigates the fabrication aspects of periodic and aperiodic assemblies of silicon nano and micro pillars in a top down approach and hierarchical ZnO nanowires on Si micro structures combining top-down and bottom-up approaches. The optical properties of the fabricated nano/micro structures are studied experimentally. Electromagnetic simulations of the optical properties of the Si nano/microstructures are also performed both for validation of the experimental observations as well as for the design of the structures for specific optical properties. Selected applications that utilize the designed specific/unique optical properties of the structures are demonstrated. Primarily, optical properties such as light propagation, localization and wavelength selective guiding have been studied for periodic and aperiodic Si nano and micro structures assemblies. These optical properties areadvantageouslyutilized to develop novel anti-reflection coatings, radial junction solar cell, surface sensitivebiosensors and color filters; and to demonstrate surface second harmonic generation (SHG).
1.1 Si micro/nano structures for antireflection and photovoltaics
Surface reflection is one of the major problems in many practical applications like solar cells, eye wear etc. In solar cells, reflection at various interfaces leads to overall decrease in efficiency. So lot of academic and industrial research is carried out to overcome this problem.
The antireflection strategy using a planar slab of index matching layer is well known. With modern day technologies different non-planar architectures are investigated for omni- directional broadband antireflection. One of the technological approaches in this direction is based on hybrid material systems, which in the broadest sense refers to inorganic-organic materials. In recent times much research interest has been on developing hybrid material systems for various optoelectronic devices. Hybrid material systems, e.g. crystalline ZnO/Si,
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combine two or more different materials with different compositional and geometrical forms to enhance overall performances and add new useful properties. One such hybrid system with hierarchical design has found many applications in photovoltaics [27], UV detection [28]
sensing [29] and as light source [30].
Si and ZnO Nanowires (NWs) hierarchical system where ZnO NWs are grown over structured Si has advantages such as large surface to volume ratio and tailored refractive index profiles. They can provide superior antireflection properties, higher light absorption [5], enhanced photoluminescence intensity [31] and multi-functionality [32]. Hierarchical ZnO NWs and micropyramid Si solar cell with reflectance of 3.2 % is reported [33]. Si-ZnO hybrid structures can be useful for many applications including antireflective coating (ARC) [33-35], solar water splitting and H2 generation [36] and sensing [37].
Another important class of nanostructured materials is the nanopillar array. Nanopillar solar cells have several advantages, like antireflective property, broadband absorption, better carrier collection etc. compared to planar bulk solar cells [38-40]. These properties can improve the solar cell’s photon to current conversion efficiency in a cost effective manner.
There are several reports where radial junction Si nanopillar or nanowires are used for solar photovoltaic power conversion [41,42]. On the other hand different doping methods are used for the p-n junction formation, including chemical vapor deposition (CVD), solid source dotation (SSD), monolayer doping (MLD) and also spin-on doping (SOD) [43-45]. Normally all these processes are carried out in conventional furnaces and need relatively high temperatures, more than 800 o C and long anneal times. These process conditions could introduce unwanted defects. On the other hand, rapid thermal annealing (RTA) process is a fast process which typically takes 60 s or less and is quite good for shallow doping which is desirable in radial PN junction formation without introduction of defects [46-48].
Combination of SOD and RTA is also reported for doping in nanopillar based devices [49,50].
In this thesis, a combination of dry and wet etch process is used to create different Si based micro and nanostructures, like nanopillars, micropillars, periodic and randomly arranged micro pyramids. A hydrothermal solution-based process is used to grow ZnO NWs on structured Si surfaces to obtain hierarchical ZnO/Si structures. Their optical properties are evaluated by total reflectivity and Raman measurements (paper I). We have also utilized the radial PN junction geometry with Si nanopillar solar cells using SOD and RTA processes. The anti-reflective properties as well as the performance of the fabricated solar cells are evaluated (paper II).
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1.2 Si nanopillar arrays for Optical Biosensing
The large demand for highly selective and sensitive biosensors for detection down to single molecule levels has led to several new techniques for biosensing [51-53]. There are several reports on methods to increase sensitivity of optical biosensors using different interrogation approaches [54, 55], materials [56, 57] and technologies [58, 59]. Nanostructure based biosensors with their unique advantages like higher surface to volume ratios and light manipulation, have been widely studied to achieve higher sensitivities [60, 61]. In biosensors based on nanopillars arrays improved sensitivities and signal to noise ratios can be obtained due to the higher effective sensing area for a given volume compared to their bulk counter parts. Nanopillars with vertical orientation can be used for many sensing applications such as fluorescence imaging [62], field enhanced fluorescence analysis [63] and enhancing signal intensity in DNA microarrays [6]. Because of the mature Si micro- and nano-fabrication technologies and superior material quality, structured Si has been increasingly used in different biosensing applications [60, 63-65].
In this thesis, we demonstrate the use of Si nanopillar arrays coated with SiO2 for biosensing. The utility of the Si nanopillar arrays together with the behavior of light in such structures is investigated both for surface and volume sensing (papers III and VI). Explicit biosensing responses were evaluated by immobilization of BSA protein and the recognition of its specific antibody (anti-BSA) (paper III; chapter 5).
1.3 Surface Second Harmonic (SHG) generation in Si nanopillar arrays
A comprehensive development of Si photonics is stalled mainly by inefficient light emission from silicon (Si) due to its indirect minimum-energy bandgap and its centrosymmetric crystalline structure hindering its use as an electro-optic modulator [3, 66]. Although many other optical functionalities of Si have been incorporated in a single platform with microelectronic devices, functions/devices such as switches, nonlinear optical devices, high speed and low power consumption are still not fully achieved [3,66]. Higher-order (> 2) nonlinearities of Si are used for different optical processes [66,67] but necessitate relatively high optical powers and are not very efficient. The problem in crystalline Si is that the second-order term of the nonlinear susceptibility tensor is forbidden in the dipole approximation due to its centro-symmetric crystal structure and the residual higher-multipole processes are too weak for practical applications [68]. However, if the inversion symmetry can be broken using inhomogeneous strain and/or by structuring (the crystal symmetry is
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naturally broken at the surface) second order nonlinear susceptibility can be induced [69-71], thus making processes like second-harmonic generation (SHG) feasible. Surface SHG was originally reported by Bloembergen and Pershan [72] and subsequently experimentally demonstrated [73]. In case of surface SHG from Si, structural and optical field discontinuities can equally contribute [74]. This property has been used to exploit SHG for probing planar surface/interfaces [75,76]. It can perhaps apply for non-planar geometries as well [77-80].
Besides surface SHG, strained sub-100 nm Si nanowires have been demonstrated to enhance SHG [81].
In this thesis, we have investigated surface SHG from hexagonal Si pillar arrays fabricated by a top down approach, including the effects of geometry of the nanopillar and modal analysis. A pump–probe setup in reflection geometry was used for SHG measurements, with the pump wavelength at 1030 nm to generate green light at 515 nm (paper IV; chapter 6).
1.4 Aperiodic Si nanopillar assemblies for color filtering applications
Color is one of the main distinctive properties of vision for living beings. In human eyes colors are distinguished by responsiveness of rods and cones to visible light of different energy/wavelength to perceive different colors. Similarly, color filters enable selection of individual colors or wavelength bands from white light, hence are key components for color imaging and display. Color pigments and dyes are the most commonly used materials in color filters. Such filters rely on material selective absorption in the visible band [82,83].There are other means of color generation, so called structural colors, based on the interaction of light with wavelength/sub-wavelength scale structures along with the material properties. A given material but with different structuring can, in principle, give different colors. This attractive solution for color filters might offer possible ways of cost reduction.But there are problems to be overcome before they become commercially attractive. For example, transmission color filters based on metallic hole arrays have been reported, but they suffer from low transmission [84]. Similarly, guided mode resonance based and photonic crystal based color filters exhibit a strong angular dependence [84-87]. Nanostructured Si has also been utilized for structural color generation but mostly in reflection mode [88].
Recently, a lot of work has been devoted to comprehend light transport as well localization effects in disordered media [89]. The results indicate several similarities between the electronic and optical wave phenomena in disordered systems for example, disorder induced localization (Andersson’s localization) [90]. Also, aperiodic structures generated in a deterministic route have attracted significant attention in optics and electronics for the
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optimization possibilities in their design to obtain specific application relevant properties [91,92]. Thus, it is worth investigating the physical mechanisms in aperiodic photonic nanostructures that can enable better performance and/or unique properties (compared to conventional as well as periodic photonic nanostructures) in photonic applications.
In this thesis, we demonstrate “stand-alone” transmission RGB color filters based on deterministically aperiodic Si nanopillar assemblies embedded in a flexible host matrix. This includes technology development, design, electromagnetic simulations and optical characterization of the color filters. The photonic properties of the assembly (pillar arrangement, pillar geometry and size) including the wavelength dependent optical absorption is used for vivid color filtering in transmission mode. Aperiodic design of Si nanopillar assemblies are implemented to get angle independent color and also transverse light localization properties are examined (paper V; chapter 7).
1.5 Thesis outline
This thesis is structured in eight chapters. Chapter 1 gives an overview of the field including different application areas and briefly discusses the relevance of the thesis work. Chapter 2 describes the most essential background and theory pertinent to this thesis. In chapter 3, the experimental procedures including the fabrication methodology, measurement and characterization procedures are elaborated. Chapter 4 describes broad-band anti-reflection and photovoltaic applications of Si nano and microstructures including hierarchical ZnO NWs on structured Si. In chapter 5 biosensing applications of Si nanopillar are reported. Surface second harmonic generation (SHG) from Si sub-micron/nanopillars is described in chapter 6.
In chapter 7 color filtering properties of aperiodic Si nanopillar assemblies is discussed. A summary of the thesis and an outlook discussing the future prospects and potential applications of the thesis work are given in chapter 8.
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2 Background and Theory
2.1 Optical properties of Silicon
Since the thesis predominantly deals with Si-based micro-/nano-structures, we give a brief account of the general optical properties of crystalline silicon at room temperature [93].
Figure (2.1a) shows the optical absorption coefficient of silicon as a function of wavelength.
Silicon being an indirect bandgap semiconductor, it has a long tail in absorption for above bandgap wavelengths. There is a sharp drop in absorption around 1100 nm, corresponding to the band edge of Si. From a photovoltaic perspective, crystalline Si is a suitable material for solar cells since it has very good absorption in terrestrial solar radiation band [94] (described in the sec. 2.2.3). However, the difficulty in crystalline Si solar cells is that for complete absorption of above band-gap solar radiation the active region is very thick ( ~ 500 µm) . In this regard, nanostructured surfaces can offer a way for enhanced absorption with less material. Appropriate nanostructuring can, in principle, provide broad-band omni-directional antireflection as well as a radial pn junction configuration for the solar cell enabling efficient carrier collection. Besides the use of absorption in structured Si for solar cells, the absorption of Si in the visible wavelength region can be also be utilized for a completely different application context such as transmission color filters. In fact, as described in chapter 7 and paper V, absorption is essential for our color filter.
The refractive index (RI) for Si is shown in fig. 2.1b. Si has an appreciably high refractive index compared to many dielectric materials such as SiO2, SiNx, and this property has made it a very good candidate for optical waveguides and other light confinement applications. The high refractive index of Si can be utilized to design miniature optical devices in the nano or
(b)
Figure 2.1. Optical properties of Si at 300K vs wavelength: (a) Absorption depth and coefficient.
(b) Real and imaginary refractive indices. (c) Reflectivity (polished substrate). The data shown in (a) and (b) are plotted from the available database [ref 93] while (c) shows the measured total reflectivity from a single-side polished Si wafer.
(a) (c)
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micron scale for controlling/molding the flow of light as well as for light trapping and localization.
The surface reflectivity is another important property that needs to be considered for device design and practical applications like solar cells, sensors etc. Since this thesis uses a top-down approach to structure Si, the reflectivity of the starting material - flat single crystalline Si substrate is often used as the reference to qualify changes in the reflectivity spectra due to structuring and/or surface coatings. Here we show the measured total reflectivity of (single-side) polished Si for normal incidence (figure (2.1c)). It is clear that reflectivity is significantly high, ~70% in UV to ~30% at 1000 nm; in the green-near IR the variation is not appreciable (40-30%). As discussed in the subsequent sections high surface reflection is a menace for high efficiency solar cells. As shown in this thesis, appropriate surface structures can be developed for broad-band suppression of reflectivity for light harvesting applications (chapter 4; papers I, II) while for other applications such as biosensors the reflectivity spectra can be modified to obtain particular peaks in reflectivity that are sensitive to refractive-index changes (Chapter 5 ; papers III and VI ).
2.2 Optical properties of Nanopillars
The interaction of light with structured materials (media) is dependent on the structure, which controls the optical field distribution and propagation of light. The structuring of materials basically changes the original geometry and invariably increases surface to volume ratios.
Often structuring or structured materials could also refer to surface topography modification of a material(s). The refractive index of structured materials also no longer remains a simple function spatially and varies along different directions depending on the structuring. The detailed arrangement of the structure may also become important in determining the optical properties of structured materials; for example, some optical properties vary considerably between periodic and aperiodic arrangements of nanopillars. However, with reference to periodic nanopillar assemblies most of the optical properties are equally observable in an aperiodic system with some variations [95].
Here, we briefly describe the properties of structured materials wherein at least two of the geometric dimensions of the individual structures are in the sub-wavelength scale. In a simple classification, the photonic nanostructures can be divided in to three different categories, namely zero dimensional particle, one dimensional nanowires placed horizontally and vertical nanopillars (horizontal and vertical orientation is defined with respect to light incidence).
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This categorization makes it easy to differentiate phenomenA responsible for a particular observation inonestructure to another depending on geometry. Referring to Fig.2.2a,b, when illuminated from the –z direction, localized optical resonances can be excited in the particles and horizontally placed NWs. Here, we have implicitly assumed the refractive index of the substrate is much lower. In horizontal structures light can be confined in the direction of light propagation (z direction) and to some extent in the sideways directions (x and/or y directions).
Whereas in vertical pillar arrays/assemblies light propagation (Fig. 2.2 c) is more dominant and their properties can be understood by understanding their wave guiding properties. Also in an assembly of pillars, be it periodic or aperiodic, the length scales, mainly period of arrays or nearest neighbor distance and filling fraction of material are important to evaluate the optical effects and also for modeling of these structures. Thus, it is not only the individual pillars but also their assembly that determines the optical properties [96]. In this thesis, we discuss the vertical nanopillars and their optical properties.
Vertical nanopillar assemblies are very important because of their higher surface to volume ratios and their ability to guide light from visible to infrared spectral range. In vertically oriented Si nanopillar arrays light can couple into a nanopillar from the top with few waveguide modes supported. In fact in subwavelength diameter wires (100 nm or less) coupling of light is possible to the allowed HE11 mode [97]. In subwavelength nanopillars, diameter dependent coupling of input plane wave at a select wavelength band of light is prominent and can lead to a dip in the reflection spectrum. In this case, with an absorbing material like Si different color in the reflected light can be produced. In finite length of pillars, only selective wavelengths may be possible to couple due to Fabry–Perot resonances in the vertical direction of pillar [98]. Another important observation of Si pillar arrays is the enhancement of light absorption. This is analyzed as coupling of the array mode to the nanostructure [99]. As explained in [99] the nanopillars can be well absorbing only for
(c)
Figure 2.2. Three of the most generic photonic nanostructures; periodic arrangement is shown for simplicity. (a) Zero dimensional nanoparticles. (b) One dimensional nanopillars or wires lying horizontally. (c) One dimensional nanopillars or wires standing vertically.
(b) (a)
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incident light that efficiently couples to selected modes which also have strong Fabry-Perot resonance between the top and bottom interfaces of the pillars, and are well guided within the silicon pillar.
In this thesis work, we have mostly studied the optical properties of Si nanopillars, related to enhanced absorption, guiding, optical field distribution in the pillars and some geometry related enhancement for different applications.
2.3 Antireflection property and solar cells
While reflection of light gives fascinating effects in nature and artificial structures it is a serious problem in applications like eyewear, solar cells and photo-detectors. For example, unwanted reflection in eyewear can be dangerous particularly while driving. Regarding solar cells, an illustration (Fig. 2.3) is given below to underline how solar cell performance is retarded by reflection or rather how it can be improved by its reduction. Figure 2.3 shows the breakup of different loses in crystalline silicon solar cell illuminated by a 100 mW normalized solar power (radiation). It include loses due to material properties, defects, cell architectures- carrier collection inefficiency, incomplete absorption, resistance and reflection. The reflections lose accounts for the percentage of useful photons lost by reflection at the top surface of the cell. Polished crystalline Si on an average has 33% of reflection, which implies that there is plenty of scope to increase efficiency of the cell by reducing surface reflection.
Figure 2.3 Loses in crystalline silicon solar cell (rough estimation)
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In the following we will elaborate different anti-reflection schemes, their evolution, merits and demerits.
The basic theory of suppressing reflection can be understood in the form of refractive index matching slab on the substrate (fig. 2.4a) [100]. For a slab of thickness mλ/4, where m is an odd number, it follows from Fresnel’s law the reflection (r) at normal incidence is
, Where ns is refractive of the substrate, nair is RI of air, and n is the RI of the slab.
The problem with this type of structure is that it can reduce reflection perfectly at a specific wavelength and only for normal angle of incidence. To make the antireflection effect broadband and omnidirectional, a gradient RI anti-reflection (AR) coating is desirable. Broad band antireflection can be obtained by a graded index layer. In a holistic way, it can be a single graded index layer; the refractive index variation is such that it matches that of air and the substrate at the top and bottom interfaces, respectively.Because of this gradual refractive index increment, out of phase reflected beams can interfere destructively for broad band
r= 0 , when n=
Figure 2.4 Different antireflection schemes that can be applied as surface coatings (a) homogeneous layer (b) inhomogeneous graded refractive index layer (c) pyramidal shape (d) Moth eye structures (surface coatings)
(a) (b)
(c) (d)
(2.1) (2.2)
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illumination, so reducing the overall reflection [101,102]. But it is quite difficult to realize for practical applications.
With the advent of surface structuring techniques, like wet chemical or plasma etching, gradient index can be created on the surface. In a good approximation, for light wavelengths much longer than the period and structure dimensions such structured surfaces (schematically illustrated in Fig. 2.4 c,d) behave like a gradient-index coating with (a depth wise) increasing or decreasing effective RI (neff ) in light propagation direction. The optical properties of this structure can be analyzed by the effective medium theory (EMT) [103]. With EMT it is possible to calculate the reflectivity of structured surfaces of various one-dimensional (1D) antireflection coatings and substrate materials. Structured or rough surfaces can be treated as a multiple layers of homogeneous refractive index film with the effective refractive index varying from substrate and air (fig. 2.5). The effective RI (neff) of the whole structure can be calculated from the packing fractions (f) of the individual structured or rough surfaces; this approach was originally developed by ellipsometric measurement techniques [104].
Figure 2.5 : Effective medium approximation (recreated after [106])
In Bruggeman’s effective medium approximation (EMA) [105] model for homogeneous mixtures of two constituent layers of refractive indices n1 and n2, we have the relation for effective index, n for whole structure as
For multiple numbers of layers, EMA gives (‘i’ is a positive integer):
EMA or EMT methods are well suited for planar graded index or one dimensional arrays of structured surface. But for two or three dimensional wavelength and subwavelength scale structures (arrays) rays optics based methods are insufficient. For such two dimensional arrays of structures, e.g. moth eyes or inverted pyramid or nanopillar/wire arrays more rigorous
(2.3)
(2.4)
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methods based on wave optics are needed to fully understand the actual effects. In many cases simple analytical solutions are not possible and numerical methods like finite difference time domain (FDTD) or finite element method (FEM) are required. In this thesis, we used the FDTD method to solve two dimensional arrays of micro and nanostructures for different applications.
In this thesis, we have implemented three dimensional antireflecting hierarchical structures consisting of Si microstructures decorated with ZnO NWs for suppressing reflection. We will discuss more on this in chapter 4.
2.4 Radial junction (pillar) solar cell
In a planar pn junction Si solar cell, contribution to output current is less probable from photo- generated carriers generated deep inside the material as they can recombine before reaching the junction. So, the carrier generation volume should be within a diffusion length of the (collecting) junction. In general, this also requires that the material be highly crystalline and free of non-radiative defects/traps. In radial junction solar cells, carriers generated deeper down in the pillar, especially, for longer wavelength photons, can reach the radial junction quickly before recombination and thereby contributing to current even for a semiconductor with a minority-carrier diffusion lengths shorter than its optical absorption depth [106]. In the radial junction geometry, the pillar can be optically long enough for absorption and at the same time the generated charge-carriers can easily reach the junction or electrode along the shorter radial distances. Also, in a radial junction pillar solar cell light can scatter or reflect from the pillar surface multiple times increasing the optical path length within the cell. Thus the pillar (arrays) geometry can contribute for much better absorption compared to the planar cell and can lead to cost effective efficiency enhancement.
2.5 Solar spectrum
It is needless to say that the solar energy is the principal source of renewable energy. The solar radiation touching the earth is more than sufficient to fulfill our energy requirements if utilized effectively. The solar radiation extends from ultraviolet to far infrared, but as shown in the Fig.2.6 the major portion of the photon flux is between 300 nm to 1800 nm [94].
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Figure 2.6 Standard Solar Spectra for terrestrial applications [ref 94]
The above AM1.5 spectrum (fig. 2.6) is for terrestrial solar cell applications and it has collective power density of 1000 W/m2 or 100 mW/cm2. For crystalline Si solar cell the effective region of the solar spectrum is from 350 to 1100 nm. The excess energy due to absorption of photons with energy much higher than the silicon band gap is dissipated in the form of phonons (carrier relaxation). Secondly, high energy photons are absorbed near the surface and the generated carriers are more prone to non-radiatively recombine at the surface (defects) and reduce photo current response at shorter wavelengths. Photons with less than the band gap of silicon are not absorbed in the cell [107].
Another important aspect of solar energy utilization is the cost-benefit ratio. Higher cost/
efficiency ratio is one of the major concerns for useful utilization of otherwise hugely abundant solar energy source. One possibility of cost reduction is to increase the efficiency with less use of materials. Less use of processed materials also has a positive implication for the environment. In this regard, nanostructured and thin film solar cells have been extensively studied for efficiency enhancement and as low cost alternatives for bulk solar cells [106].
2.6 Nanopillars and Optical biosensing
In a generic sense, in an optical biosensor, when an input, such as a biomolecule comes in the range of interaction of the sensor it causes a change in the characteristics of the sensor, for example in its reflection or transmission spectrum or output power. The sensitivity of the sensor to a particular species depends on the magnitude of the change in the output signals it returns in a repeatable manner. Fig. 2.7a shows a generic biosensor.
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In dielectric nano pillar arrays, selective reflection can be taken to advantage for biosensing.
Depending on its lateral dimensions, nanopillars can guide light with a large optical power outside, which can facilitate interaction with the surrounding environment. Thus, control over the optical field distribution is an important aspect that determines the performance of the nanopillar biosensor [108,109]. Another important aspect is the higher surface to volume ratio as this can lead to a sensitive miniaturized biosensor.
In a reflectivity based biosensor, the working principle is based on effective wavelength shifts in the interference fringes in the reflected light spectrum caused by the change of refractive index due to biomolecule attachment. Hence, such sensors are in general called refractive index sensors. In a nanopillar biosensor, binding of biomolecules on the pillar surface induces changes in the effective refractive index of the nanopillar medium. In the presence of biomolecules, wavelength shifts in the interference fringes in the reflection spectrum are observed (fig. 2.7 b). Normally, the wavelength shifts are proportional to the amount of the biomolecule present [110,111].
In a refractive index sensor, the sensitivity is influenced by the portion of the optical power that interacts with the sample. For resonant refractive index sensors, such as Fabry-Perot sensors, effective refractive index variations due to biomolecules (and their different concentrations) Δnsleads to a spectral shift Δλ which can be expressed as,
Where η is the portion of light intensity that interacts with the sample and neff is the effective refractive index experienced by the resonant modes present in the system [112,113].
Since the biosensor system consists of a optical readout section the spectral shift (i.e., the sensitivity) alone does not determine its sensitivity but also the readout accuracy. So it is
Figure 2.7 (a) Schematic of generic biosensor (b) Nanopillar biosensor
(a) (b)
(2.5)
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important to define the sensor resolution as the smallest possible spectral shift it can detect with selectivity and precision. Hence, the spectral resolution of the readout and its noise parameters are important [114]. The lowest amount of biomolecules a sensor can detect accurately is called the limit of detection (DL) and is related to the sensitivity (S) and resolution (R) of the sensor as:
In most of the refractive index biosensors, the concentration of biomolecules and the observed spectral shift is not always linear and can saturate at high biomolecule concentrations. The estimation of DL is discussed further in paper III.
2.7 Second harmonic generation and Si nanopillars
In nonlinear optics, light induced non-linear changes in the optical properties of materials are studied. To observe nonlinear phenomenon sufficiently intense light, usually using high power laser light, is required to modify the optical properties of a material system. The induced polarization P is described by the following equation [115] :
𝑃𝑃 = 𝜖𝜖0 [𝜒𝜒 (1)E + 𝜒𝜒 (2)E 2 + 𝜒𝜒 (3) E3 + 𝜒𝜒 (4) E4 ⋯ ] ≡ 𝑃𝑃 (1) +𝑃𝑃 (2) + 𝑃𝑃 (3) + 𝑃𝑃 (4)…….
Where E is the electric field of light, 𝜖𝜖0 is the permittivity of free space. 𝜒𝜒 (2) and 𝜒𝜒 (3) are the second-order and third-order nonlinear optical susceptibilities and so on.
As the nonlinear response of materials depends on the crystallographic orientation and direction of the electric field (E), it can more conveniently be described by a tensor relationship.
The second-order nonlinear polarization for SHG in tensor form can be written as follows:
𝑃𝑃𝑖𝑖(2)
= 𝜖𝜖0∑j,k=𝑥𝑥,𝑦𝑦,𝑧𝑧𝜒𝜒𝑖𝑖𝑗𝑗𝑘𝑘(2)𝐸𝐸𝑗𝑗𝐸𝐸𝑘𝑘
Where 𝜒𝜒𝑖𝑖𝑗𝑗𝑘𝑘 (2)
is the 2nd order nonlinear susceptibility and subscripts represents three orthogonal directions. To represent nonlinear polarization in all possible directions, 𝜒𝜒𝑖𝑖𝑗𝑗𝑘𝑘(2)
can be written as a tensor with 27 components, but effectively only 18 and hence in a more simplified version the tensor is written with respect to the nonlinear optical coefficient tensor, dij as [116],
(2.6)
(2.7)
(2.8)
